Transcript Chapter 11

Section 11-2
Goodness of Fit
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
1
Key Concept
Sample data consist of observed frequency
counts arranged in a single row
(called a one-way frequency table).
We will test the claim that the observed
frequency counts agree with some claimed
distribution.
In other words, there is a good fit of the
observed data with the claimed distribution.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
2
Definition
A goodness-of-fit test is used to
test the hypothesis that an
observed frequency distribution
fits some claimed distribution.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
3
Goodness-of-Fit Test
Notation
O
(letter, not number) represents the Observed
frequency of an outcome
E
represents the Expected frequency of an outcome
k
represents the number of different categories or
outcomes
n
represents the total number of trials
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
4
Goodness-of-Fit Test
Requirements
1. The data have been randomly selected.
2. For each category, the expected frequency
is at least 5.
(There is no requirement on the observed
frequency for each category.)
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
5
Goodness-of-Fit
Test Statistic
(O  E )
x 
E
2
2
2
x
is pronounced “chi-square”
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
6
Goodness-of-Fit
Critical Values
1. Found in Table A- 4 using k – 1 degrees
of freedom, where k = number of
categories.
2. Goodness-of-fit hypothesis tests are
always right-tailed.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
7
Goodness-of-Fit
P-Values
P-values are typically provided by
computer software, or a range of Pvalues can be found from Table A-4.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
8
Expected Frequencies
If all expected frequencies are equal:
n
E
k
the sum of all observed frequencies
divided by the number of categories
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
9
Expected Frequencies
If expected frequencies are
not all equal:
E  np
Each expected frequency is found by
multiplying the sum of all observed
frequencies by the probability for the
category.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
10
Goodness-of-Fit Test
A close agreement between observed and
expected values will lead to a small value of x 2
and a large P-value.
A large disagreement between observed and
2
x
expected values will lead to a large value of
and a small P-value.
2
A significantly large value of x will cause a
rejection of the null hypothesis of no difference
between the observed and the expected.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
11
Goodness-of-Fit Test
“If the P is low, the null must go.”
(If the P-value is small, reject the null
hypothesis that the distribution is as
claimed.)
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
12
2
Relationships Among the x Test Statistic,
P-Value, and Goodness-of-Fit
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
Figure 11-2
13
Example:
Data Set 1 in Appendix B includes weights
from 40 randomly selected adult males and
40 randomly selected adult females. Those
weights were obtained as part of the
National Health Examination Survey. When
obtaining weights of subjects, it is
extremely important to actually weigh
individuals instead of asking them to report
their weights. By analyzing the last digits of
weights, researchers can verify that weights
were obtained through actual
measurements instead of being reported.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
14
Example:
When people report weights, they typically
round to a whole number, so reported
weights tend to have many last digits
consisting of 0. In contrast, if people are
actually weighed with a scale having
precision to the nearest 0.1 pound, the
weights tend to have last digits that are
uniformly distributed, with 0, 1, 2, … , 9 all
occurring with roughly the same
frequencies. Table 11-2 shows the
frequency distribution of the last digits from
80 weights listed in Data Set 1 in Appendix
B.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
15
Example:
(For example, the weight of 201.5 lb has a
last digit of 5, and this is one of the data
values included in Table 11-2.)
Test the claim that the sample is from a
population of weights in which the last
digits do not occur with the same
frequency. Based on the results, what can
we conclude about the procedure used to
obtain the weights?
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
16
Example:
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
17
Example:
Requirements are satisfied: randomly
selected subjects, frequency counts,
expected frequency is 8(  5)
Step 1: at least one of the probabilities
p0 , p1... p9 , is different from the others
Step 2: at least one of the probabilities are the
same:
p0  p1  p2  p3  p4  p5  p6  p7  p8  p9
Step 3: null hypothesis contains equality
H0 : p0  p1  p2  p3  p4  p5  p6  p7  p8  p9
H1 : At least one probability is different
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
18
Example:
Step 4: no significance specified, use   0.05
Step 5: testing whether a uniform distribution
so use goodness-of-fit test: x 2
Step 6: see the next slide for the computation
of the x 2 test statistic. The test statistic
x 2  11.250 , using   0.05 and k  1  9
degrees of freedom, the critical value is
2
x  16.919
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
19
Example:
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
20
Example:
Step 7:
Because the
test statistic
does not fall
in the
critical
region,
there is not
sufficient
evidence to
reject the
null
hypothesis.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
21
Example:
Step 8: There is not sufficient evidence to
support the claim that the last digits
do not occur with the same relative
frequency.
This goodness-of-fit test suggests that the last
digits provide a reasonably good fit with the
claimed distribution of equally likely
frequencies. Instead of asking the subjects
how much they weigh, it appears that their
weights were actually measured as they
should have been.
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
.
22
Goodness-of-fit by TI-83/84
• Press STAT and select EDIT
• Enter Observed frequencies into the list L1
• Enter Expected frequencies into the list L2
then the procedure differs for TI-83 and TI-84:
• In TI-84: Press STAT, select TESTS
scroll down to
c2 GOF-Test , press
ENTER
Type in: Observed: L1
Expected: L2
df: number of degrees of freedom
Press on Calculate
read the test statistic c2 =...
and the P-value p=…
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
23
Goodness-of-fit by TI-83/84
• In TI-83: Clear screen and type
(L1 - L2 )2 ÷ L2 → L3
(the key STO produces the arrow → )
Then press 2nd and STAT, select MATH
scroll down to sum( press ENTER
at prompt sum( type L3 then ) and press ENTER
This gives you the c2 test statistic
For the P-value use the function c2 CDF
from DISTR menu and at prompt c2 CDF(
type test_statistic,9999,df)
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
24